3.2
Analytical Model for the AoI Evaluation
We focus on periodic one-hop message exchange, with nodes sending messages of given length L with period Tmsg. New messages are accepted by the MAC layer entity
of a node as long as it is idle. If the node is busy, the arriving new message is stored in a buffer. Further arriving messages are overwritten so that only the latest message is taken care of by the MAC layer, as soon as the previous message has been sent out on the channel. This setting is consistent with the periodic issuing of the beacon messages, each carrying an update of the vehicle information. Given this setting, only the latest update is worth being transmitted.
The model describes the generic node operation with a renewal process. Let us consider a tagged node A sending a message at time tk, k∈ Z. We let Yk= tk−tk−1. At
equilibrium, we can assume Yk∼ Y . Since the times {tk}k∈Zare regeneration points
for the sending process, the sequence of intervals{Yk} forms a renewal process6.
We can distinguish two cases according to whether the contention plus transmission time is less than the message inter-arrival time (top picture in Figure 3.2) or not (bottom plot on Figure 3.2).
Then
Y = B2+ max{0, Tmsg− B1} (3.1)
where B is a random variable defined as the time elapsing from the moment when the MAC layer takes in charge a PDU until it eventually sends it out on the radio channel. B is the sum of the transmission time T (including the overhead) and of
6At least, this is true under the simplifying assumption of independence of the stations’ states.
T
msgC
T
Y
B
2B
1T
msgC
T
Y
B
2B
1Figure 3.2 – The time interval elapsed from the moment when the message is
3.2 Analytical Model for the AoI Evaluation 31
the time C spent on counting until the back-off counter hits 0. If the channel is sensed busy, the counter is frozen until the channel activity terminates. Otherwise, the counter is decremented after a back-off time slot of durationδ. Let C denote the count-down time, defined as the sum of a number of "slot" times, each slot lasting eitherδ, the IEEE 802.11p back-off slot duration, or T, which is the time required to complete a MAC PDU transmission, including PHY and MAC overhead, and the ensuing DIFS7. Then
C= N
X
j=1
X(j) (3.2)
where N is a discrete random variable with uniform distribution over[0, W0− 1], W0
being the base contention window size of IEEE 802.11p, and X(j) are i.i.d. random variables with the same distribution as X defined by
X = δ w.p. 1 − b T w.p. b (3.3)
Here b is the probability that the tagged node senses the channel busy.
In the evaluation of the statistics of X and hence of C, we must account for the fact that only partial sensing is realized in general. In other words, while some neighbor N1of the tagged node A is transmitting, and hence the tagged node freezes
its count-down state, some other neighbor N2of A could start its own transmission, in case N2is out of carrier sensing range of N1, i.e., N1is hidden to N2. The resulting
effect as seen by the tagged node A is that its freezing time lasts more than T . This ‘expanded‘ duration of the activity sensed on the channel by A depends on the maximum number of nodes that can start transmitting independently of one another, i.e., that do not sense one another. Let nMISdenote the cardinality of the MIS around A. Once the transmission starts, up to nMIS− 1 more transmissions could start with
random phases. By assuming independence and uniform probability distribution of the relative phasing in[0, T], it can be found easily that the time T is replaced by T(2 − 1/nMIS). The expansion factor ψ = 2 − 1/nMIS≥ 1 of the activity time
reduces to 1 when nMIS= 1, i.e., all neighbors of A do sense each other. A proxy of the number nMISthat is easier to calculate is ˆn= c + ν(1 − c), where ν is the
number of neighbors of A and c is the clustering coefficient of A. This is simply a linear interpolation between nMIS= 1 when c = 1 and nMIS= ν when c = 0. The
clustering coefficient of a graph node A is the ratio of the number of links among theν neighbors of A divided by the maximum number of such links, i.e., ν(ν − 1)/2. Given the adjacency matrix A of an undirected graph, the clustering coefficient of node i is ci = `i/[ν(ν − 1)/2], where `i can be found as the i-th element of the 7SIFS and ACK times are not included, since MAC PDUs for beaconing are sent in broadcast, hence no
32 3.2 Analytical Model for the AoI Evaluation
diagonal of the matrix A3/2. In the definition of Xi, the random variable X at node i, we therefore substitute T with T ˆψi, where ˆψi= 2 − 1/ˆni= 2 −ci+νi1(1−ci), with ci=
2`i
νi(νi−1).
We have the following identities for the first two moments of Bi= T + Ci
E[Bi] = T + W0− 1 2 E[Xi] σ2 Bi = W02− 1 12 (E[Xi]) 2+W0− 1 2 σ 2 Xi where E[Xi] = δ(1 − bi) + T ˆψibi σ2 Xi= (T ˆψi− δ) 2b i(1 − bi)
The first two moments of Yiare found by considering all realizationsβk of the
random variable Bi and the relevant probabilities, i.e., pi(k) ≡ P (Bi = βk). By
definition: E[ max{0, Tmsg− Bi} γ ] = W0−1 X k=0 pi(k) max{0, Tmsg− βk}γ
forγ ≥ 1, and then Equation (3.1) yields
E[Yi] = E[Bi] + E[max{0, Tmsg− Bi}] Var(Yi) = Var(Bi) + Var max{0, Tmsg− Bi}
where, for k= 0, . . . , W0− 1., we have
pi(k) = P (Bi= βk) = 1 W0 W0−1−k X m=0 m+ k k bki(1 − bi)m (3.4) and βk= T + W0δ + k(T ˆψi− δ) (3.5)
The probability that the i-th node attempts a transmission on the channel is τi= τ0
E[Bi]
E[Yi]
(3.6) whereτ0is the probability of attempting a transmission in a saturated CSMA/CA network, when binary exponential backoff is not used and only the basic contention window size is used. Hence,τ0 = 2/(1 + W0), with W0 = 15, according to the IEEE 802.11p standard. Note that nodes do not operate necessarily in saturation,
3.2 Analytical Model for the AoI Evaluation 33
since they are requested to send one message every Tmsg. As long as Bi< Tmsgnode icompletes contention and message transmission before the next message is ready to send. This is the typical case for standard message periods (between 100 ms and 1000 ms), given that the contention time ranges between few ms and several tens of ms typically.
There remains to characterize the probability b. Let us introduce a subscript i for the tagged node. Let ai jdenote the entry(i, j) of the adjacency matrix A of the carrier sensinggraph of the nodes. In words, ai j= 1 if and only if node j can receive
(detect) the signal emitted by node i. Since the radio channel is reciprocal, we can assume that A is symmetric. In this model, we assume that the carrier sensing matrix A is given (see Section 3.4).
Asτjis the probability that node j is found transmitting, the probability that
a neighbor node j of i is not transmitting is 1− τjaji. We adopt the common
independence assumption, whereby the states of the competing nodes in the CSMA network are assumed to be independent of one another. Then, the probability that node i senses an idle channel, i.e., that all its neighbors are silent, is8
1− bi= n
Y
j=1
(1 − τjaji) (3.7)
where n is the number of nodes in the network, hence the size of the adjacency matrix.
Summing up, theτi’s can be found by solving a system of non-linear equations
made up of Equations (3.3), (3.6) and (3.7). If we writeτ ≡ [τ1τ2 . . . τn], the
equation system can be written in a compact form asτ = F(τ). The function F(·) is continuous and maps the unit hypercube into itself. Hence, Brouwer’s theorem guarantees that there exists a fixed point.
Once the transmission probabilitiesτiare computed, we can find the conditional
probability of success, Ps(i, j), of the event that node j receives a message from node i, given that i transmits the message. This amounts to node i transmitting and: (i) none of the neighbors of j being active at the same time; (ii) node j not transmitting as well. We can divide the neighbors of j into two sets:
Ai, j the set of neighbors of j that are also neighbors of i;
Bi, j the set of neighbors of j that are not neighbors of i.
The nodes belonging to the first set are synchronized by the activity of i, while the other nodes are not. Therefore, the transmission probability for node k∈ Ai, jis τk. Nodes inBi, jare outside the communication range of i, hence they are hidden
with respect to i. We assume they are completely de-synchronized with i, hence
8Note that we define a
34 3.2 Analytical Model for the AoI Evaluation
node k∈ Bi, jcan start transmitting in any slot time of durationδ with probability δ/E[Yk]. The vulnerability interval of the message sent by node i to node j comprises m≡ 2T /δ − 1 slot times. Therefore
Ps(i, j) = (1 − τj) Y k∈Ai, j 1− τkak j Y k∈Bi, j 1− δ E[Yk] ak j m
for all j6= i. The time Zi j to deliver a new message from i to j is given by
Zi j= Ni j X
r=1
Yi(r) (3.8)
where Yi(r) ∼ Yiare the times between successive transmission attempts of node i, Yiis given in Equation (3.1), and Ni j is the number of attempts required to make a
successful message transfer from i to j. Assuming that successive attempt outcomes are independent of one another, Ni j has a geometric probability distribution, i.e.
P(Ni j= h) = Ps(i, j)[1 − Ps(i, j)]h−1 (3.9) for h ≥ 1. The AoI at node j for messages coming from i equals t − ti j(k) for t∈ [ti j(k), ti j(k) + Zi j), where ti j(k) is the time of arrival of the k-th message from ito j.
The mean value of the AoI from i to j, Hi j, is akin to the mean remaining service
time in a queue, i.e.
E[Hi j] = E[Z2 i j] 2E[Zi j] (3.10) It is
E[Zi j2] = E[Ni j(Ni j− 1)](E[Yi])2+ E[Ni j]E[Yi2]
=2[1 − Ps(i, j)] Ps(i, j)2 (E[Yi]) 2+ 1 Ps(i, j)E[Y 2 i ] =2− Ps(i, j) Ps(i, j)2 (E[Yi])2+ 1 Ps(i, j) σ2 Yi E[Zi j] = 1 Ps(i, j) E[Yi]
The expressions above allow to compute the mean AoI of messages flowing from ito j. The AoI at j can be obtained by averaging over all neighbor nodes of j, if any. If j is isolated, it receives no message actually, so AoI is meaningless. Besides this